A and $C$ are immiscible in the liquid state. Liquid $A-B$ solution is known to be ideal.

A crucible containing $3$ moles $A,0\mathrm{.}5$ moles $B$ and $2$ moles $C$ is placed in a furnace at $1500C.$ The system at equilibrium is as shown in the sketch below. Analysis of a sample taken from the $C-$layer shows that $x_B=0\mathrm{.}1.$

$($a$)$ Calculate the activity of $B$ in the $C-B$ system for this composition at $1500C.(15$ p$)$

$($b$)$ Calculate the activity coefficient of $B$ in the $C-B$ system for this composition at $1500C.(5$ p$)$